3.329 \(\int \frac{x^8}{\left (a+b x^3\right )^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3}{3 b^2} \]

[Out]

x^3/(3*b^2) - a^2/(3*b^3*(a + b*x^3)) - (2*a*Log[a + b*x^3])/(3*b^3)

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Rubi [A]  time = 0.0739244, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^2}{3 b^3 \left (a+b x^3\right )}-\frac{2 a \log \left (a+b x^3\right )}{3 b^3}+\frac{x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^3)^2,x]

[Out]

x^3/(3*b^2) - a^2/(3*b^3*(a + b*x^3)) - (2*a*Log[a + b*x^3])/(3*b^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{3 b^{3} \left (a + b x^{3}\right )} - \frac{2 a \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{\int ^{x^{3}} \frac{1}{b^{2}}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**3+a)**2,x)

[Out]

-a**2/(3*b**3*(a + b*x**3)) - 2*a*log(a + b*x**3)/(3*b**3) + Integral(b**(-2), (
x, x**3))/3

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Mathematica [A]  time = 0.0322786, size = 38, normalized size = 0.83 \[ \frac{-\frac{a^2}{a+b x^3}-2 a \log \left (a+b x^3\right )+b x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^3)^2,x]

[Out]

(b*x^3 - a^2/(a + b*x^3) - 2*a*Log[a + b*x^3])/(3*b^3)

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Maple [A]  time = 0.007, size = 41, normalized size = 0.9 \[{\frac{{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{2}}{3\,{b}^{3} \left ( b{x}^{3}+a \right ) }}-{\frac{2\,a\ln \left ( b{x}^{3}+a \right ) }{3\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a)^2,x)

[Out]

1/3*x^3/b^2-1/3*a^2/b^3/(b*x^3+a)-2/3*a*ln(b*x^3+a)/b^3

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Maxima [A]  time = 1.43601, size = 58, normalized size = 1.26 \[ -\frac{a^{2}}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} + \frac{x^{3}}{3 \, b^{2}} - \frac{2 \, a \log \left (b x^{3} + a\right )}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a)^2,x, algorithm="maxima")

[Out]

-1/3*a^2/(b^4*x^3 + a*b^3) + 1/3*x^3/b^2 - 2/3*a*log(b*x^3 + a)/b^3

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Fricas [A]  time = 0.211139, size = 76, normalized size = 1.65 \[ \frac{b^{2} x^{6} + a b x^{3} - a^{2} - 2 \,{\left (a b x^{3} + a^{2}\right )} \log \left (b x^{3} + a\right )}{3 \,{\left (b^{4} x^{3} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a)^2,x, algorithm="fricas")

[Out]

1/3*(b^2*x^6 + a*b*x^3 - a^2 - 2*(a*b*x^3 + a^2)*log(b*x^3 + a))/(b^4*x^3 + a*b^
3)

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Sympy [A]  time = 1.76444, size = 42, normalized size = 0.91 \[ - \frac{a^{2}}{3 a b^{3} + 3 b^{4} x^{3}} - \frac{2 a \log{\left (a + b x^{3} \right )}}{3 b^{3}} + \frac{x^{3}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**3+a)**2,x)

[Out]

-a**2/(3*a*b**3 + 3*b**4*x**3) - 2*a*log(a + b*x**3)/(3*b**3) + x**3/(3*b**2)

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GIAC/XCAS [A]  time = 0.222052, size = 66, normalized size = 1.43 \[ \frac{x^{3}}{3 \, b^{2}} - \frac{2 \, a{\rm ln}\left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{3}} + \frac{2 \, a b x^{3} + a^{2}}{3 \,{\left (b x^{3} + a\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a)^2,x, algorithm="giac")

[Out]

1/3*x^3/b^2 - 2/3*a*ln(abs(b*x^3 + a))/b^3 + 1/3*(2*a*b*x^3 + a^2)/((b*x^3 + a)*
b^3)